Abstract
In the present article, we prove the \(L^\infty \)-stability of discontinuous or supercritical shock waves which appear in a model system of radiating gases if the shock strength is greater than a certain critical value. The author has recently shown (SIAM J. Math. Anal. (2014), 2136–2159.) that all subcritical shock waves are stable to small perturbations while the critical shock wave blows up the first order derivative in a finite time if certain types of perturbations are added whatever small the perturbations may be. In the supercritical case, we show that the convection contributes to recover the stability by virtue of discontinuity in the asymptotic state compensating the insufficient smoothing effect of radiation.
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Acknowledgements
The author wishes to express his sincere gratitude to the financial support from the Alliance for Breakthrough between Mathematics and Sciences (ABMS) supported by the Core Research for Evolutional Science and Technology (CREST) of the Japan Science and Technology Agency (JST), and to the Japan Society for the Promotion of Science (JSPS) for Grant-in-Aid for Young Scientists (B), No. 26800076. This work was done through a research project ‘Study on Mathematical Fluid Mechanics’ at Research Institute for Science and Engineering, Waseda University, and through the Japanese-German Graduate Externship (Mathematical Fluid Dynamics) supported by JSPS.
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Ohnawa, M. (2016). \(L^\infty \)-Stability of Discontinuous Traveling Waves in a Radiating Gas Model. In: Shibata, Y., Suzuki, Y. (eds) Mathematical Fluid Dynamics, Present and Future. Springer Proceedings in Mathematics & Statistics, vol 183. Springer, Tokyo. https://doi.org/10.1007/978-4-431-56457-7_20
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DOI: https://doi.org/10.1007/978-4-431-56457-7_20
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